Foundations of Garside Theory

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Patrick Dehornoy

with François Digne, Eddy Godelle, Daan Krammer, and Jean Michel

A publication of the European Mathematical Society

This text is a monograph on algebra, with connections to geometry
and low-dimensional topology. It mainly involves groups, monoids, and
categories, and aims to provide a unified treatment for those
situations in which one can find distinguished decompositions by
iteratively extracting a maximal fragment lying in a prescribed family.
Initiated in 1969 by F. A. Garside in the case of Artin's braid groups,
this approach led to interesting results in a number of
cases, the central notion being what the authors call a Garside family.
The study is far from complete, and the purpose of this
book is to present the current state of the theory and to
invite further research.

The book has two parts: In Part A, the bases of a general theory,
including many easy examples, are developed. In Part B, various more
sophisticated examples are specifically addressed.

To make the content accessible to a wide audience of
nonspecialists, the book's exposition is essentially self-contained and very
few prerequisites are needed. In particular, it should be easy to use
this as a textbook both for Garside theory and for the
more specialized topics investigated in Part B: Artin–Tits
groups, Deligne–Lusztig varieties, groups of algebraic laws,
ordered groups, and structure groups of set-theoretic solutions of the
Yang–Baxter equation. The first part of the book can be used as
the basis for a graduate or advanced undergraduate course.

A publication of the European Mathematical Society (EMS).
Distributed within the Americas by the American Mathematical Society.

This text is a monograph on algebra, with connections to geometry
and low-dimensional topology. It mainly involves groups, monoids, and
categories, and aims to provide a unified treatment for those
situations in which one can find distinguished decompositions by
iteratively extracting a maximal fragment lying in a prescribed family.
Initiated in 1969 by F. A. Garside in the case of Artin's braid groups,
this approach led to interesting results in a number of
cases, the central notion being what the authors call a Garside family.
The study is far from complete, and the purpose of this
book is to present the current state of the theory and to
invite further research.

The book has two parts: In Part A, the bases of a general theory,
including many easy examples, are developed. In Part B, various more
sophisticated examples are specifically addressed.

To make the content accessible to a wide audience of
nonspecialists, the book's exposition is essentially self-contained and very
few prerequisites are needed. In particular, it should be easy to use
this as a textbook both for Garside theory and for the
more specialized topics investigated in Part B: Artin–Tits
groups, Deligne–Lusztig varieties, groups of algebraic laws,
ordered groups, and structure groups of set-theoretic solutions of the
Yang–Baxter equation. The first part of the book can be used as
the basis for a graduate or advanced undergraduate course.

A publication of the European Mathematical Society (EMS).
Distributed within the Americas by the American Mathematical Society.